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Theorem prdstset 17455
Description: Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
prdsbas.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
prdsbas.b 𝐡 = (Baseβ€˜π‘ƒ)
prdsbas.i (πœ‘ β†’ dom 𝑅 = 𝐼)
prdstset.l 𝑂 = (TopSetβ€˜π‘ƒ)
Assertion
Ref Expression
prdstset (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))

Proof of Theorem prdstset
Dummy variables π‘Ž 𝑐 𝑑 𝑒 𝑓 𝑔 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2728 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 prdsbas.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝑉)
5 prdsbas.r . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘Š)
6 prdsbas.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
71, 4, 5, 6, 3prdsbas 17446 . . 3 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
8 eqid 2728 . . . 4 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
91, 4, 5, 6, 3, 8prdsplusg 17447 . . 3 (πœ‘ β†’ (+gβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
10 eqid 2728 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
111, 4, 5, 6, 3, 10prdsmulr 17448 . . 3 (πœ‘ β†’ (.rβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
12 eqid 2728 . . . 4 ( ·𝑠 β€˜π‘ƒ) = ( ·𝑠 β€˜π‘ƒ)
131, 4, 5, 6, 3, 2, 12prdsvsca 17449 . . 3 (πœ‘ β†’ ( ·𝑠 β€˜π‘ƒ) = (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
14 eqidd 2729 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))))
15 eqidd 2729 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
16 eqid 2728 . . . 4 (leβ€˜π‘ƒ) = (leβ€˜π‘ƒ)
171, 4, 5, 6, 3, 16prdsle 17451 . . 3 (πœ‘ β†’ (leβ€˜π‘ƒ) = {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))})
18 eqid 2728 . . . 4 (distβ€˜π‘ƒ) = (distβ€˜π‘ƒ)
191, 4, 5, 6, 3, 18prdsds 17453 . . 3 (πœ‘ β†’ (distβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )))
20 eqidd 2729 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
21 eqidd 2729 . . 3 (πœ‘ β†’ (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))) = (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))))
221, 2, 3, 7, 9, 11, 13, 14, 15, 17, 19, 20, 21, 4, 5prdsval 17444 . 2 (πœ‘ β†’ 𝑃 = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
23 prdstset.l . 2 𝑂 = (TopSetβ€˜π‘ƒ)
24 tsetid 17341 . 2 TopSet = Slot (TopSetβ€˜ndx)
25 fvexd 6917 . 2 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) ∈ V)
26 snsstp1 4824 . . . 4 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩}
27 ssun1 4174 . . . 4 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βŠ† ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})
2826, 27sstri 3991 . . 3 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})
29 ssun2 4175 . . 3 ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}) βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
3028, 29sstri 3991 . 2 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
3122, 23, 24, 25, 30prdsbaslem 17442 1 (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βˆͺ cun 3947  {csn 4632  {cpr 4634  {ctp 4636  βŸ¨cop 4638   ↦ cmpt 5235   Γ— cxp 5680  dom cdm 5682   ∘ ccom 5686  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7997  2nd c2nd 7998  Xcixp 8922  ndxcnx 17169  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  Β·π‘–cip 17245  TopSetcts 17246  lecple 17247  distcds 17249  Hom chom 17251  compcco 17252  TopOpenctopn 17410  βˆtcpt 17427   Ξ£g cgsu 17429  Xscprds 17434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-hom 17264  df-cco 17265  df-prds 17436
This theorem is referenced by:  prdshom  17456  prdsco  17457  prdstopn  23552  prdstps  23553
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